Production (L4)

Main source: NS Ch 6 (parts)

Coverage

NS: Ch 6: Production
Production function (at least the basic idea)
Returns to scale (!)
Basic cost concepts (!)
‘Cost minimising input choice’ (understand key principle of ‘bang-for-the-buck’ equalisation)
Cost curves (!)

… to help understand the key concepts in following chunks

Key goals of this chunk

Understand how economists consider a firm
… and depict a firm’s production function
… and how a firm can ‘trade off one input for another’ in producing a particular output
… and why the production function may have increasing returns to scale or diminishing returns to scale

Think about: How does the firm’s production choice differ from a consumer’s consumption choice?

Who cares about the firm’s/firms’/national/global production function?

How will you earn a living in the future? Production may be a function of capital (esp ‘high-tech’), materials, and various types/skill-levels of labour. We’ll note that in a competitive market each input, including workers, is paid based on their marginal contribution to output, given the firms’ optimising choice of inputs. If production functions (or costs of various inputs) change ‘your kind of labour’ may become less productive (per hour) as part of this process, meaning wages must fall. If they fall low enough people may not want to work. If these are ‘sticky’ or legally prevented from falling firms may reduce employment of ‘your kind of labour.’ 2020 US presidential candidate Andrew Yang is making this sort of argument, and he claims the best response is a tax-funded ‘universal basic income’. Do you think he is correct?

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Urgent question: Brexit

Trade with Europe may default to WTO terms

\(\rightarrow\) Very large tariffs on some goods, ‘non-tariff barriers’ on others

UK (and EU) firms: Unknown impact on input prices (as well as demand curves, competition, etc.)

How will this affect the production process?

\(\rightarrow\) Consumer prices? Wages? Investment returns?

Can ‘GE models’ help firms plan and reoptimise?

The firm and the production function

How do we define it? What does it do?
What’s its goal?

The Firm’s Production function

Production function: mathematical relationship between inputs and outputs.

\[q = f( K, L, M, ...)\]

\[q = f(K, L)\]

e.g., \(q =2L^{1/3}K^{2/3}\)

Given choice of ‘amount to produce’, firm tries to produce it at minimum cost

(like consumer maximizing utility st budget constraints)

Marginal product

Marginal product: Additional output from adding +1 unit of an input, holding other inputs constant.

Marginal Product of Labour—\(MP_L\): Slope of production function in units of labour (holding capital etc constant)

\[MP_L=\frac{\partial}{\partial L}f(K,L)\]

Similarly for Marginal Product of Capital (\(MP_K\))

A standard assumption: diminishing MP of each input

Output, marginal & average product (curves)

Isoquant maps, rate of technical substitution (RTS)

Similar to consumer’s indifference curves, but quantity is observable

Marginal rate of technical substitution (RTS): Amount one input can be reduced when +1 unit of another input is added, holding output constant

RTS = - slope of the isoquant

|(change in capital) / (change in labor)|

\[RTS = MP_L/MP_K\]

RTS is the ratio of marginal products

Makes sense:

If MPL large I can give up much K, bc gaining L adds a lot of production
If MPK large I can’t give up much K to get more L, bc reducing K reduces output a lot
Diminishing RTS

Cost minimizing input choice

A particular quantity \(q\) can typically be produced with a variety of different input combinations

Which combination will the firm choose?

\[q = f(K,L,...)\]

Main point: Whatever q it wants to produce, firm uses the minimum cost combination of inputs!

…chooses inputs to get the best ‘bang for the buck’;

\(\rightarrow\) where input mix optimal, each input yields same marginal product per £

Consider a combo of K and L, and the output this yields.

From here, firm can ‘substitute capital for labour’ at rate \(RTS(K,L)= MP_L/MP_K\) (& hold production constant)

If firm uses both K & L and chooses optimally \(\rightarrow\)

Set \(RTS(K,L)\) equal to the input price ratio of these inputs (\(w/r\))

\(\rightarrow\) ‘Same bang for the buck at optimal choices \(K^*\) and \(L^*\)’ i.e.,

\[\frac{MP_K(K^*,L^*)}{r}= \frac{MP_L(K^*,L^*)}{w}\]

Aside, for intuition and story-telling

If markets for labour and capital are (perfectly) competitive prices of inputs & outputs adjust so that:

‘bang for the buck’ (in revenue) is the same for all inputs and for all production processes
Inputs (workers, owners of capital) in every industry will be paid based on their (marginal) productivity …

Advanced question

Back to our ‘replaced by AI’ motivation, a simple model. Suppose:

One product in the economy with ‘constant returns to scale Cobb-Douglas production function:’

\[q = L^{a}K^{1-a}\]

This implies ‘optimising firms will spend a share \(\alpha\) on labour’.

Now suppose technological changes imply \(a\) decreases. What do you think will happen to wage rates, presuming a ‘fixed supply of labour’?

Summing up

Summing up: Optimisation (given a production function and input prices)…

yields a (minimum) cost for every output \(q\) a firm chooses to produce.

\(\rightarrow\) We will be able to construct a cost function

Returns to scale

Are bigger firms always more efficient? Do things get cheaper to produce the more we produce?

Returns to scale: The rate at which output increases in response to a proportional increase in all inputs.

Constant returns to scale (CRS): If inputs increase by a factor of X, output increases by a factor equal to X.

E.g., doubling all inputs (labor, capital, land, etc) means exactly doubling all outputs

Increasing returns to scale (IRS): If inputs increase by a factor of X, output increases by a factor greater than X.

Decreasing returns to scale (DRS): If inputs increase by a factor of X, output increases by a factor less than X.

Arguments/reasons for scale (dis)economies

IRS

Fixed costs (incorporation, buildings, management, planning, R&D) spread over more units
Should always be able to at least ‘double everything’ and produce twice as much? (so at least CRS)
Scale allows specialisation

Arguments for DRS

Limited resources in (relevant) economy; costs begin to rise
Managerial issues and coordination problems, bigger ‘centre’ to lobby for favours. See ‘theories of the firm’
Harder to give incentives to top manager/CEO?

Interesting case: ‘Minimum efficient scale’ (MES) production

E.g., rental costs/maintenance of 3D printer: 10,000 per year (no matter how much is printed) Each printer can print up to 100 artificial heads per year If produce \(<\) 100 heads/year, cost/head not minimised.

Here MES is 100 \(\rightarrow\) You should produce in multiples of 100, but could be CRS for each multiple of 100

Computing… If you know the production function, how do you know if the ‘returns to scale’ are increasing or decreasing?

Slide in a constant \(\alpha>1\) next to each input, simplify, and compare to the original production

E.g.:

\[Q(L,K) = L^{1/4}K^{1/2}\]

\[Q(\alpha L, \alpha K) = (\alpha L)^{1/4}(\alpha K)^{1/2}\] \[=\alpha^{1/4}\alpha^{1/2}L^{1/4}K^{1/2}=\alpha^{3/4}L^{1/4}K^{1/2}=\alpha^{3/4}Q(L,K)\]

\(\rightarrow\) So if we increase inputs by \(\alpha\) here, we increase output by \(\alpha^{3/4}<\alpha\), so DRS everywhere for this production function.

Costs

Main source: NS Ch 7 (just a few parts)

Key goals of this chunk

Characterise and contrast different types of costs; how they should enter into firms’ decisions (and life decisions!)
Continue to depict a firm’s cost-minimising input choice

and its expansion path

Types of costs (‘Basic cost concepts’)

Fixed costs (FC): Costs that must be regularly incurred to remain in business (i.e., for any level of output), but that do not vary with the level of output

Variable costs (VC): Costs that increase with the quantity produced.

Sunk costs: Costs that have been incurred in the past that can never be recovered.

Sunk costs should not enter into any economic decisions.

FC from previous years are sunk costs; FC for future years are not.

Again: Economic profits and cost minimisation; 2-input model

Labour costs - wage rates \(w\)
Capital costs - rental rate \(v\)

Total costs \(= TC = wL + vK\)

Economic profit = \(\pi\) = Total Revenues - Total Costs

\[\pi = Pq-wL-vK = P\times f(K,L)-wL-vK\]

(P: price of good)

Cost minimizing input choice, expansion path, ratio condition

Choose point where RTS = ratio of input prices

RTS = (wage rate/rental rate) = w/v
Same ratio of ‘marginal productivity/price’ for each input used

Which point on this curve will minimize cost? . . . The one on the lowest ‘isocost line’ … similar to budget line for consumers

Intuition

\[RTS = MP_L/MP_K = w/v\]

Cost-minimisation for each level of output \(\rightarrow\) firm’s expansion path

Total cost curve: from cost of inputs along expansion path

Average and marginal cost curves

Read at home:

Average and marginal cost curves
… with differing returns to scale: Increasing, decreasing, constant, optimal scale

Skip: “Short and long run” costs for the firm

Average cost: Cost per unit of output

\[AC=TC/q\]

TC = ‘summed’ marginal costs and fixed costs
AC: ‘average marginal cost’ + FC/q

Marginal cost: Incremental cost of last unit produced

I.e., additional cost of producing one more unit of output
MC: Slope of (optimising) TC curve at a point

Shape of marginal cost curve depends on production function

Constant returns to scale: constant MC (and no FC)
Decreasing returns to scale: increasing MC
Increasing returns to scale: decreasing MC (and/or constant FC)

On estimating cost/production functions

Want to estimate product, firm, and industry production functions
Government (regulators etc), forecasters, strategists

Anti-trust, regulating natural monopolies, macro-growth issues and aggregate production, importance of human-K, impact of trade deals (winners/losers), impact of min. wage and labour laws, business strategy and competition, reacting to anticipated market changes…

Production functions \(\Leftrightarrow\) cost functions

Amount that can be produced with any set of inputs \(\Leftrightarrow\) (Minimum) cost of producing any output

Difficult to estimate these: Lack of publicly available data, lack of exogenous variation in input choices